The constructed solution \((u,B)\in C_tH^{\widetilde{\beta }}_x \times C_tH^{\widetilde{\beta }}_x\)  in [1]},
where \(\widetilde{\beta }>0\) , can be approximated by
the relaxation solutions \((u_q,B_q,\mathring{R}_{q}^u,\mathring{R}_{q}^B)\)  to (REF )
with \(\nu _1=\nu _2=0\) ,
which satisfy the inductive estimates that for any \(q\ge 0\) 
(see [1]})
\(& \Vert {u_q}\Vert _{C_tL_{x}^{2}} \le 1-\delta ^{\frac{1}{2}}_q,\quad \Vert _{C_tL_{x}^{2}} \le 1-\delta ^{\frac{1}{2}}_q, \\& \Vert {u_q}\Vert _{C_{t,x}^{1}} \le \lambda _{q}^{2},\quad \Vert _{C_{t,x}^{1}} \le \lambda _{q}^{2}, \\& \Vert \mathring{R}_q^u\Vert _{C_tL^{1}_{x}} \le c_u \delta _{q+1},\quad \Vert \mathring{R}_q^B\Vert _{C_tL^{1}_{x}} \le c_B \delta _{q+1}, \\& \left\Vert u_{q+1}-u_{q}\right\Vert _{C_tL^{2}_{x}} \le \delta _{q+1}^{\frac{1}{2}}, \quad \left\Vert B_{q+1}-B_{q}\right\Vert _{C_tL^{2}_{x}} \le \delta _{q+1}^{\frac{1}{2}},\) 
